In the bootstrap approach to integrable quantum field theories in the (1+1)-dimensional Minkowski space, one conjectures the two-particle S-matrix and tries to study local observables. The massless sine-Gordon model is conjectured to be equivalent to the Thirring model, and its breather-breather S-matrix components (where the first breather corresponds to the scalar field of the sine-Gordon model) are closed under fusion. Yet, the residues of the poles in this breather-breather S-matrix have wrong signs and cannot be considered as a separate model. We find CDD factors which adjust the signs, so that the breather-breather S-matrix alone satisfies reasonable assumptions. Then we propose candidates for observables in wedge-shaped regions and prove their commutativity in the weak sense.